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V. V. Skripachev 

Translation of "Ustoychivost ' laminamogo pogranichnogo 
sloya na def ormiruyemoy poverkhnosti membrannogo 
tipa," Zhumal Prikladnoy Mekhaniki i 
Teknicheskoy Fiziki, No. 6, 
1969,' pp. 52-56 

t[ , ' ,' ^'i^*;i L'^ m^i g'{ M ears 

v^^ L-ii ^^^j ft™, I ^ i §^ i^ 

Pi r^ 

WASHINGTON, D.C. 205^ J^J^ 1970 

NASA TT F- 12,969 


V. V, Skripachev 

, ABSTRACT. The stability of the Blasius flow over a diaphragm-type 
surface, the physical characteristics of which are constsint along 
the length, is examined « 

Attempts have been made to provide a theoretical explanation of the effect /32* 
boundary deformation has on the position of the point in the boundary layer at 
which stability is lost. These attempts are associated with Kramer's successful 
experiments Cl, 2] in sheathing models with flexible coatings. A. I. Korotkin [j] 
escaained the stability of a plane lasiinar boundary layer over an elastic sxjrf ace 
on the assumption that there is a linear connection between pressure disturbance 
and norsial surface deformation. Benjamin C^] and Laadahl [53 investigated the 
stability of a laminar boundary layer over a diaphragm-type surface on the assump- 
tion that the physical characteristics of the surface depend on the wavelength of 
the disturbing flow. 

The stability of the Blasius floiif over a diaphragm-type surface, the physical 
characteristics of vSiich are constant along the length, will be exaained in what 

. We will take it that when there are no disturbances the surface of the plate 
coincides with the half -plane x > 0, y = (Figure l). Let us suppose that -cer- 
tain disturbances have taken place in the flow at a predetermined moment. Let us 
investigate the stability of the flow with respect to these disturbances. 



Figure 1 

Numbers in the margin indicate pagination in the foreign text. 

Let U, V (V <^ U) be the cofflponents of the velocity of the Blasius flow along 

the X and y axes, respectively, p be the pressure, v be the kinematic raodulus of 

viscosity, and p be the fluid density, VJe will take it that the velocities of 

the disturbances, u', v% and the pressure disturbance, p', are small in the 

sense that terms that are quadratic with respect to the disturbances can be 

ignored. Let us introduce the stream function, '^\ for the disturbing flow 

in the form 

ip' = q) (y) exp [i a{x—ct)] (l) 

the Tidiile asstiming that the real part of equation (l) is taken. The number. 
Of, is a real magnitude, linked tdth the wavelength of the disturbing flow by the 
relationship or = 2n/\. The phase velocity c = c + ic. is a complex magnitude. 
The sign of the imaginary part, c, tells whether the disturbance ydll increase 
(c. > O), or be damped (c. < O). Dimensionless magnitudes are used in equation 
(l), as well as in what follows. The velocity Ug at the outer limit of the 
boundary layer is taken as the velocity scale, and the thickness of the boundary 

is taken as the length scale. 

The neutral curve c. =0, separating the region of rising disturbances from 
the region of damping disturbances, is of particular interest. The Reynolds 
niiffiber for loss of stability is detenained by the shape of this curve. The 
neutral stability curve is constructed from the solution of the Orr-Soamerfield /53 
eqization for the amplitude cp of the stresua fimctioa for a disturbing flow L6] 

{U-c) (9" - a?<p) - £^*<P = -^Tfi (^''' - ^''''P" + "^'P^ (2 ) 

•i?=i£i^ U = 2y~5y^+Qy^-2y^ 


The boundary conditions for equation (2) express the. conditions for disttirb- 
ance , damping at infinity and the adhesion conditions. The conditions at infinity 
are in the form [6] 

cp'-jra(p = 0, 1 (p 1< oo ^ (3) 

The adhesion conditions express the equality of the velocity of a surface 
element and a liquid particle adjacent to .the surface (figure l) 

i^(^-^^ . = u\x + I. -ri) + u' {X + 1, Tl) 


.^^ (^- ^y = F (a; + I, Ti) + y' (a; + 1, n) • 


Let us put 

g (a;, i) = 116^* '*"'='>. Ti(a;,i) = T]ie' 

_ ~,,eia(a!-ci) 


Substituting the equality at (5) in (^)? expanding the right sides of the 
latter in a Taylor series, and taking the siaallness of the deforsaations, and the 
velocity V, into consideration, we obtain 


% — — 3— (6) 

It .will be convenient, in subsequent computations, to introduce the normal, 
Y , and. the tangential, X , yielding of the flow with respect to the traveling 
wave. The noitnal (tangential) yielding is determined with sign correctness by the 
ratio of the normal (tangential) velocity to the pressure disturbance, 
p' = p_ exp [ia (x - ct)], that is 

■Y - - ^(«' + g.^) + "'('^ + g-'n) ■ 
which can be written with first order of infinitesimals correctness as 

The asjplitude p^ of the pressure disturbance can be found from the linearized 
equations for the motion of a viscous fluid in projections on the x and y axes, 

p,= ^[<p"'(0)-aV(0)3 + '=<P'(0) + ^'(°^'P^°^ (8) 



The identity of equations (8) suid (9.) follovjs from equation (2). 

Introduced in similar fashion are the tangential, Y-p, and normal , Y.-, 
yieldings of a defonaable surface vdth respect to the traveling wave 

. 1 se .. „ _. i_ii' ■ 

and can be written vdth first order of infinitesimals correctness in the form /5^ 

^12 jr ' -^^ ~ ^?r" (10) 

The equality 

y„ = F„. X. == v.. 


jq — Yix, Xq — y^2 

yields the boundary conditions on the deformable surface. 

The computations show that the tangential yielding has little effect on the 
position of the point at which stability is lost, and it can be taken as equsQ. 
to zero. 

Let us, in order to determine the normal yielding, Y__, viiich is dependent 
on Tj-, consider the motion of a diaphragm, element (ilgure 2).. 

™^ = -/-/A, + ra.-<i$" ■■■: 


Figure 2 

Here si^ is the mass of a unit of diaphragm area, T^ is the surface tension 
occurring j>er unit width of the diaphragm, and k^ is the stiffness factor. The 
asterisks denote dimensional magnitudes. 

Let us find the "n-i/p-. ratio from equation (12), vdth equation (5) taken into 
consideration. When this ratio is substituted in equation (lO) we obtain 

^11 = 


ma? (Co* — c* — cid/'ma) 



Co^ = Co^n.+ "°' " -' 

-rs- , Com =T — , fflo^ = — = kJ'R^ 

An approximate solution of equation (2) can be given in the form 

In this equation § is the "nonviscous. " solution, satisfying the equation 

(J7 - c) (O" - d'O) - i7" O == 

■ - - (15) 

and <p^ is the approximate "viscous" solution of equation (2) satisfying 

equation [6] 

d^'a 'a 


, 6 =• (aRU,')-'f' I 


Here Y is the value of y for which U = c. ' 

The solutions 'of § and cp™ satisfy the boundary conditions at (3). The 
boundary conditions at (ll), and the condition of nontriviality of the solution 
lead to the characteristic equation linking the magnitudes a^CjE with the para- 
meters of the deformable surface. Before writing this equation, let us simplify 
the expression for the pressure amplitude, p., contained in (11 ), ignoring the 

small magnitude terms. In accordance with equations (8) and (l4), we can write 

p^ = O"'(0)-a"I''(0) -,. ■^^a"'^0)-o^>',(q). + ,^/ (Q)' + U' (0) cp (0) 


The first term in the right side of equation (17 ) can be ignored because 
the change in the nonviscous solution is slow. This term is exactly equal to 
zero in the case of the Blasius flow, as follows from eqtiation (15) after 
differentiation with respect to y. The sum of the third and fourth tesnas in 
the right side of equation (17 ) « in accordance with equations (6) and (lO), 
equals cY ■,-)P^i and it too can be ignored. In order to make further simplifi- 
cations in equation (l7)» let us find cp,"' (O) from equation (l6) by term 
integration with respect to y 

(P3"'(0) = -Jai?2/o£^oV(0) 



Equation (l8) yieldsb,'" (0)|>|cp^« (0)|. Therefore, taking the fact that 

U.'ii« U' (0), y U '« c, and using equations (7) and (ll), our final finding is 
c o c 

. i>x (0) = .27' (0)0 (0) + cO' (0) ^^^^ 

An identical expression can also be obtained by the transformation of 

equation (9)» .and it will be correct to ^dthia the S ' terms. The arguments 

cited above confirm Landahl's assertion [5] that a linearized equation of motion 

in the projection on the y axis yields a more accurate expression for pressure 

disturbance than does a linearized eqiiation of motion in the projection on the 

X axis. 

Using the expression obtained for p. , we can write the characteristic 
equation in the form 

{Zu IV (0)CD (0) + cO' (0)1 - iaO (0)} W (0)cp3 (0) + ccps' (0)] = 

, . = -ia93 (0) [J7' (0)(D (0) -h c(D' (0)] \ (20) 

Let us simplify equation (20). Let us introduce the following notations: 

£^^=-i^(.). ,^^^-,^:; (21) 

i?* {z)=^u + iv + . ^^ 

Here F(z) is a Tietjens function^ tables of vSiich are contained in [63. 
Equation (20) when expressed in teims of the notations in (21) will, after 
uncomplicated transf ozonations, appear in the form 

V (0) Yn ' 

The function §, in terms of ^ich we can express ti + iv, is determined by 

the solution of eqiiation (15). Presenting this solution in the form of a series 


in terras of powers of a , and limiting ourselves to the principal terms, we 

obtain [6] 

Let us substitute the value for Y__ from equation (13) ia eqiaation (22), 
and let us isolate the real and the imaginary pairts. We find that 

■ p :!._,• mU'(0){coyc~c) ■■.. 

^«_ ■ - ■ U'(0)d ■ .■.. ^^^^ 

■ * ""^ . aid^ + w'a'CcoV* — c)'] 


F * and F,* are the real and the imaginairy parts of the function F*(z) 

Let us note that the link between the pressure and the deformation sometimes 
is given in the form [33 

that is, in accordance with equation (lO), it is taken that 

and not considered as a concrete form of adeformable surface. With equation (13) 
in mind, it is not difficult to see that for the model of a deformable surface 
adopted here 

that is, K- and 6 depend on the physical parameters of the disturbance wave and 


on the parameters of the deformable surface. 























Figure 3 

Based on the foregoing, the construction of the neutral stability curve for 
fixed parameters of a deformable surface can be carried out in the following 
sequence. Find F^* and F.* in the tables for each z. Solve equations (2^) and 
(21) for or amd c. Compute the corresponding R number by solving the corresponding 
equation at (21). Corresponding to the K number for loss of stability are 

2 = 3.21, F * = 0.58, and F * = 1.49. 

1 r 

Figure 3 shows the results of the S number computations for loss of stability. 
Curve 1 depicts the dependence of 3 = R/E. on the parameter for the mass 

\ ' ^m^ml "^^^ °o = °*'^5' ^0) =>-56-10"^, d = 0.1, k^^ = l,8.1o\ The E. 


number corresponds to the R number when k = k , , differing little from the R 

ra ml ° 

number for a rigid surface. Curve 2 is for the dependence of 3 on h = k A" 

0) QJ I 



when \ = 0.4, c^ = 0,75, d = 0.1, k . = 7.4rl0 . It should be pointed out 



that the selection of the values for c and d in these computations was more 

or less arbitraiy. 

Submitted 13 May 1969. 


1. Kramer, M. 0., Boundary layer stabilization by' distributed damping, 

J. Amer. Soc. Naval Engrs. , I96O, Vol. 72, No. 1. 

2. Kramer, M. 0., Boundary layer stabilization by distributed damping. 

Naval Engrs. J. , I962, Vol. 74, No. 2. 

3. Korotkin, A. I., Ustoychivost ' laminarnogo pogranichnogo sloya v 

neszhimayemoy zhidkosti na uprugoy pbverkhnosti [Stability of the 
Laminar Boundary Layer in an Incompressible Fluid Over an Elastic 
Surface]. Izv. AN SSSR, MZhG, I966, No. 3. 

4. Benjamin T. B. , Effects of a flexible boundary on hydrodynamic 

stability, J. Fluid Mech. , I960, Vol. 9i Pt. 4. 

5. Landahl M. T. , On the stability of a laminar incompressible boundary 

layer over a flexible surface, J. Fluid Mech. , 1962, Vol. 13, pt. 4. 

6. Lin' Tszya-uzyao, Teoriya gidrodinamicheskoy ustoychivosti [Hydrodynamic 

Stability Theoiry]. Moscow. Foreign Literature Press, 1958. 

Translated for National Aeronautics and Space Administration under contract 
No. NASw-2038 by Translation Consultants, Ltd., 944 South Wakefield Street, 
Arlington, Virginia 22204.